Is there a modern browser for iOS 10.3.3 capable of supporting ES6+ Javascript?
Solution 1:
From what little I know about ECMAScript Safari on iOS (which all browsers must use for rendering web pages, as @Mark pointed out in his comment) some things are supported but many are not. And your guess is as good as anyone's as to why that is the case (not to mention off-topic here).
So you are likely out of luck for full ECMAScrip support on iOS unless you want to go the jailbreak route. We can't help with that and I don't know if there is a browser for jailbroken iOS devices that does support ECMAScript.
Solution 2:
As far as I know, not currently.
According to this ES6 compatibility table, the only iOS browser supporting this is Safari in iOS 11 (which supports 99% of the ES6 implementation).
It's not clear though how often the above table is updated, or whether it would include other browsers when they offer support for ES6+ on iOS (I note they only list Safari for iOS and no other browser).
Since there are many browsers available for iOS and most of them are regularly updated, it would be worth downloading a few of them and updating regularly to see if any of them support ES6+ in future.
As a minimum I would continue to keep an eye on Chrome, Firefox and even Microsoft Edge (which even supports iOS 9) in case they offer support. However, it would be worth keeping an eye on Opera and Dolphin as well.
[EDIT]
Just a minor clarification. The reason I am recommending that you download and keep an eye on other browsers is that their developers may opt to make them still compatible with earlier devices, even though at their core they're still using the Webkit engine. In other words, while the version of Safari using the latest Webkit only works on iOS 11 or above, there's nothing stopping another browser using the latest Webkit being compatible with iOS 10, etc.